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D > (5 (/» in CIA 2\9 UC-NRLF $B Sm Mfl5 IN MEMORIAM FLORIAN CAJORl Heath's Mathematical Monographs | r, '' Mf-ral editorship of | - Webster Wells, S.B. : .;;^.ru- of Mathematics ■- -c Massachusetts Institute of Technology SUPPLEMENTARY ALGEBRA BY R. L. SHORT Heath h Co., Piibhs^^ New York Chicago Number lo Price^ Ten Cents The Ideal Geometry' Must have Clear and concise types of formal dem- onstration* Many safeguards against illogical and inaccurate proof. Numerous carefully graded, original problems. Must Afford ample opportunity for origin- ality of statement and phraseology without permitting inaccuracy* Call into play the inventive powers without opening the way for loose demonstration. Wells's Essentials of Geometry meets all of these demands. Half leathery Plane and Solid, J gQ pages. Price $1.2^^. Plane^ 'j^ cents. Solid, 75 ce;ns. D. C. H EATH & CO., Publishers BOSTON NEW YORK. CHICAGO STUDENTS IN ATTENDANCE AT STATE UNIVERSITIES OF THE CENTRAL WEST, 1885-1903. i s & s S 1 3 s S s i s s g § "1 s s § 700 ^>-/->^ 700 600 / 600 600 / / 500 400 / / 400 300 / / »0 200 li / / / 200 100 / / / 100 3000 / / / 3000 900 / ^ f 900 800 f / / " / 800 700 / / / 700 600 J -^ \. / / J 600 600 / / /' 1 600 400 / 1 ■7 400 300 y 1 / / / 800 200 / / // r -■' 200 100 / / ■y 100 2000 / / // 2000 900 / / .^ y 900 800 / / / / 800 700 / / // / 700 800 / / ^'.-^ / / / 60O 500 /' / / x- ><-. / A. 600 400 1^ / // -/ J yi 7 400 300 y^ / / i. / J / 800 200 / / -h / / f / 200 100 / / , i > / / ...... y J ^7 r 100 1000 / / z / J ^ / 1000 900 / A ■r / p, * y / 900 800 / // i\ / 800 700 / ^ / • J ^^ / ^ 700 600 J A ^ -/) ^ / 600 500 / +- / / / J> 600 400 / y rtSi ■:^ A 400 300 —y ^ ::^' / 800 $ s £0 88 s 1 s 8 g c> s s s s s i s s 1 s Jllinoia J«ebruka>- 'WUcoiuiiL- lowa. -^ Michigan Indiana - Kansas •— Minneiota . Ohio SUPPLEMENTARY ALGEBRA BY R. L. [short BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 190S CAJORl Copyright, 1905, By D. C. Heath & Co. QA2, S55 PREFACE The large number of requests coming from teachers for supplementary work in algebra, espe- cially such work as cannot be profitably intro- duced into a text, has led me to collect such material into a monograph, hoping by this means to furnish the teacher with methods and supple- mentary work by which he may brighten up the algebra review. As far as possible, illustrations have been drawn directly from calculus and mechanics, — this being especially true in the problems for reduction. Almost without exception, such algebraic forms are common to calculus work. For a large part of this list of algebraic forms in calculus, I am indebted to Miss Marion B. White, Instructor in Mathematics in the University of Illinois. No attempt has been made to demonstrate the theory hinted at in graphical work. Such treatment would involve a knowledge of higher mathematics. The graph of the growth of state universities was made by members of a freshman class of the University of Illinois. R. L. S. SUPPLEMENTARY ALGEBRA. Graphs. 1. It is impossible to locate absolutely a point in a plane. All measurements are purely relative, and all positions in a plane or in space are likewise relative. Since a plane is infinite in length and infinite in breadth, it is necessary to have some fixed form from which one can take measurements. For this form, assumed fixed in a plane, Descartes (1596-1650) chose two intersecting lines as a co- ordinate system. Such a system of coordinates has since his time been called Cartesian. It will best suit our purpose to choose lines intersecting at right angles. ' 2. The Point. If we take any point My its posi- tion is determined by the length of the lines QM= X and PM = y, the directions of which are paral- lel to the intersecting lines C^Xand (9F(Fig. i). The values x = a and y = b will thus determine a point. The unit of length can be arbitrarily chosen, but when once fixed remains the same throughout the problem under discussion. QM=x and PM =y, we call the coordinates of the point M. x, measured parallel to the line OX^ is called the abscissa, y, measured parallel to the Hne OY, is 5 6 Supplementary Algebra. the ordinate. OX and V are the coordinate axes. OX is the axis of x^ also called the axis of abscissas. (9 Fis the axis of y, also called the axis of ordinates. (9, the point of intersection, is called the origin. Fig. I. A plane has an infinity of points in its length, also an infinity of points in its breadth. The num- ber of points in a plane being thus, oo^, or twofold infinite, two measurements are necessary to locate a point in a plane. For example, x=2 holds for any point on the line AB (Fig. 2). But if in addition we demand that J = 3, the point is fully determined by the intersection of the lines AB and CD^ any point on CD satisfying the equation j?' = 3. 3. The Line. Examine one of the simplest con- ditions in X and J, for example ;ir -j- j = 6. In this equation, when values are assigned to x^ we get a Graphs. Y A c D a' X X Y ^ B Fig. 2. value of J/ for every such value of x. When x== o, j = 6;jr=i,j=S;;ir=2, j = 4;;ir=3, j=3; ;ir = 5, J = I ; etc., giving an infinite number of values of x and j which satisfy the equation. There is, then, no definite solution. Laying off these values on a pair of axes, as shown in paragraph 2, we see that the points satis- fying this equation lie on the line AB (Fig. 3). It is readily seen that there might be confusion as to the direction from the origin in which the measure- ments should be taken. This is avoided by a simple convention in signs. Negative values of :r are measured to the left of the 7-axis, positive to the right. In like manner, negative values of j/ are measured downward from the ;r-axis, positive values upward. The regions XO F, VOX^, X'O Y\ Y^OXy are spoken of as the first, second, third, and fourth quadrants respectively. (See Fig. 2.) 8 Supplementary Algebra. By plotting other equations of the first degree in two variables (two unknown quantities), it will be seen that such an equation always represents a Fig. 3. straight line. This line AB (Fig. 3) is called the graph of X +j/ = 6 and is the locus of all the points satisfying that equation. 4. Now plot two simultaneous equations of the first degree on the same axes, e.g. x-\-y = 6 and 2x — 3^=— 3 (Fig. 4), and we see at once that the coordinates of the point of intersection have the same values as the x and y of the algebraic solution of the equations. This is, then, a geometric or graphical reason why there is but one solution to a pair of simulta- neous equations of the first degree in two unknowns. A simple algebraic proof will be given in the next Graphs, 9 article. Hereafter an equation of the first degree in two variables will be spoken of as a linear equation. \ Y \ \ D \ ,-/ y \ y A > ? \ / A \ J/ y \ .^ c ^ ^5 J Fig. 4. 5. Algebraic Proof of Art. 4. Two simultaneous equations of the first degree cannot have two sets of values for x and y. Given the two equations ax-\-by=^c, (i) ex -{-fy = k, (2) Eliminating J/, {af — eh)x = cf — bh, (3) Let x^ and x^ be the roots of (3), different in value. Substituting these roots, we have {af— eb)x^ = cf—bky {of— eb^x^ = cf— bk, (of— eb){x^ — x^) - o. lo Supplementary Algebra. But x^^x^^y ,\af=eb, or - = — , which is im- possible. "^ In general, the plotting of two graphs on the same axes will determine all the real solutions of the two equations, each point of intersection of the graphs corresponding to a value of x and y satisfying both equations. 6. It is well to introduce the subject of graphs by the use of concrete problems which depend on two conditions and which can be solved without mention of the word equation. Prof. F. E. Nipher, Washington University, St. Louis, proposes the following : "A person wishing a number of copies of a letter made, went to a typewriter and learned that the cost would be, for mimeograph work : $1.00 for 100 copies, 1^2.00 for 200 copies, $3.00 for 300 copies, ^.00 for 400 copies, and so on. " He then went to a printer and was made the following terms : $2.50 for 100 copies, ;^3.oo for 200 copies, $3.50 for 300 copies, $4.00 for 400 copies, and so on, a rise of 50 cents for each hundred. Plotting the data of (i) and (2) on the same axes, we have : (I) (2) Graphs II "The vertical axis being chosen for the price- units, the horizontal axis for the number of copies. "Any point on line (i) will determine the price for a certain number of mimeograph copies. Any point on line (2) determines the price and corre- sponding number of copies of printer's work.'* Numerous lessons can be drawn from this prob- lem. One is that for less than 400 copies, it is less expensive to patronize the mimeographer. For 400 copies, it does not matter which party is patronized. For no copies from the mimeographer, one pays nothing. How about the cost of no copies from the printer } Why 1 Any problem involving two related conditions, as in line (i), depends on an equation. The graph of an equation will always answer any question one wishes to propound concerning the conditions of the problem. The graph offers an excellent scheme for the 12 Supplementary Algebra. presentation of the solution of indeterminate equations for positive integers — a subject always hazy in the minds of beginners. Example. Solve 3;ir4-47=22 for positive inte- gers. Plotting the equation, we have Y s V N X ^ s, o s s. 2 3 k k X 6 N s si 4 3i I Fig. 6. We see that the line crosses the corner of a square only when x= 2 and x=6. For all other integral values of ;r, y is fractional. The only posi- tive integral solutions are, therefore, x=2, 7 = 4; x = 6, j/=i. This corresponds to the algebraic result. 7. The Curve. We will now plot a curve of higher degree than the first. Take, for example, ;r2 -f j;/2 = 25, or ^2 _- 25 — ;r2. Giving x either posi- tive or negative values, will give positive and negative values for j/. When x=± i, 7 = ± V24 ; ^=±2, j/ = ±V2i; x=±^yj/=±4; etc., until •^= ± S> after which value we find that y becomes imaginary. Plotting these values, we find that our Graphs. 13 equation is represented by a circle whose radius is 5 (Fig. 7). Again, substituting values of x in x^-\-y^^2^y we have the sets of points in Fig. 7, which, when joined by a smooth curve, give a circle of radius 5 for the locus of the equation, r X y ±0 ±5 ±1 ±V24 ±2 ±V2T ±3 ±4 ±4 ±3 ±5 ±0 Fig. 7. 2. Examples. s. 6. 3- ^= I. 25 16 7. It will be noted that the graph of every equation in two variables of higher degree than the first is a curve. Solve by means of graphs, and compare results with the algebraic solution. X' 25 16 16 +4=1. yl- 16. y = x^ — x— 12. 14 Supplementary Algebra. 7^ JJ 1 t 1- i - : t V t - r \ t - j^t -^ 3 X J' -3 — 20 — 2 O — I 6 O 4 I o i -i 2 o 3 lO Fig. 8. Graphs. ir 2. x^+J/^= 13, ;t:2~j^2_. j^^ It is seen in Example i that the line cuts the curve in two points. It is, in general, true that a straight line cuts a quadratic curve in two points only. These points may be real and different, real and coincident, or both imaginary. 8. Examine the curve f = x^ — x^ — 4x + 4. Al- lowing X to vary, we obtain the set of values, x=- 3,7=-20; x=:-2yj/ = o; x=-i,j/ = 6; ;r=o,j/ = 4, etc. (Fig. 8). Plotting our curve, we see that it has something of an inverted S-form, that it has two bends or turning points ; also, that it cuts the ;r-axis in three points; namely, when;ir= i, ;»r= — 2, ;r= 2. These are the roots of the equation just plotted. In general, there are as many roots to an equa- tion as there are units in the degree of the curve. Now these roots or intersections may not all be real. Take, for example, the curve j/ = x^-\-x^ — 2;ir+ 12 (see Fig. 9). We see that the curve has the same general form as the curve in Fig. 6, but that it crosses the axis in only one point; namely, x= —3. An algebraic solution gives imaginary results for the other two roots, x= I + V— 3 and x= i — V— 3. That is, these two crossing points are imaginary. i6 Supplementary Algebra. X y -4 -28 -3 o — 2 12 — I 14 O 12 1 "M "M I 12 2 20 H- Fig. 9. Graphs. 17 This shows also, to some extent, why imaginary points always go in pairs, for it is impossible to draw a curve so that it will cross the axis in only one imaginary point, or, in fact, in any odd number of imaginary points. Imaginary intersections may also occur in simul- taneous equations. In general, two simultaneous equations of the second degree intersect in four points, but the curves may be so situated that two, or perhaps all four, of the intersections may be imaginary. Take the curves. 7- Sz 7- k -N Fig. 10. Plotting these curves, we see that the first repre- sents a circle with its center at the point x= 2, y = 6, the radius of which is 2 ; the second repre- 1 8 Supplementary Algebra. sents a circle having its center at the origin and a radius equal to 2 ; also that these circles do not intersect. Solving the equations algebraically, we get for the solutions, x = • 5 \ both of which are imaginary. (See Fig. lo.) The^-values are also imaginary. If we replace that first circle by (^~l)2 + (j- 1)2 = 4, we get real points for the algebraic solution, I ± V/ I T V/ X = , y = — Fig. II. Plotting the curves (Fig. ii), we see that there are also two real intersections. Note that the graphical method gives real inter- sections only. The reason is that our axes are Graphs. 19 chosen in a plane of real points, and imaginary points are in an entirely different field. Imagi- naries are plotted by what is known as the Argand diagram, but are not within the scope of the present paper. Example. Plot }-^=i, and x^-\'r^=i6 25 9 on the same pair of axes. The result should give four rea/ intersections. ■ ■ I ' I I I I T' Fig. 12. 9. The Absolute Term. Determine what effect the absolute term has on the curve. Examine the curve j/=x^—x—i2 (Fig. 12). The curve turns at the point x=^y j/= — 12^, and when;ir=o, j^= — 12. Replace 12 by another constant, say 6. The curve now becomes j^=x^—x—6 (Fig. 13), which turns 20 Supplementary Algebra. at the point x=^y j^ = — 6|, and crosses the j^-axis at —6. Replace 6 by o. y^x^-'X — o (Fig. 14). This curve turns at ;r = |, 7 = — J, and crosses the axes at the origin. Fig. 14. Graphs. 21 In these three figures the curves are of the same form, but are situated differently with respect to the axes. It is seen, then, that the absolute term has to do with the position of the curve, but has nothing to do with its form. Moreover, a change in the absolute term lowers or raises the curve, and does not shift it to the right or left. 1 i t 1 \ X I I / 1 I / 1 \ 1 1 \ \ 1 1 \ 1 1 < 1 1 \ 1 \ \ / \ / \ \J J X 1 ' V V r ■ Fig. 15. ID. Change in the Size of Roots. We learn in the theory of equations that the substitution of x-\-n for Xy where n is positive, gives another equation, the roots of which are each less by n than the roots of the original equation. Or, if ;r — ;^ be substituted for ;r, where n is positive, each root of the new equation will be greater by n than 22 Supplementary Algebra, the roots of the original equation. In 4: = 6, if ;r + 2 is substituted for ;r, we get x= 4, 3. root less by 2 than the former equation. Plot the curve y = x^ — x — 12 (Fig. 1 5), the con- tinuous curve. Substitute x-\- 2 for x in the above equation, and we have j/ = (x+ 2j^ — {x+ 2)— 12, or7 = ;r2 + 3;i;— 10; plot on the same axes — the dotted curve. The transformed curve crosses the axis at x=2f x== — S, while the original curve crosses at -;ir = 4, ;ir= — 3, the new roots each being 2 less than the old ones, while the form of the curve is unchanged. We see, then, that a substitution of x±n for x shifts the curve, parallel to itself, to the right or left, and does not change its form. The shape of the curve must therefore depend on the coefficients and the exponents of x and j/. Plot f = x^, y=z(^x—if, y=={x—2fy on the same axes. It is seen that the curves are the same curve, merely shifted each time to the left. All curves of the form j/ = (x — of can be reduced to the form j/ = x^hy shifting the axes. Plot J/ = x^, J/ = x^, J/ = x^, y = ;tr2«, on the same axes where a is positive, and note the similarity (but not an identity) in the form of the curves. II. Plotjv = - for both positive and negative values of x. We see that as x grows smaller, y constantly increases, and that as x approaches zero, y becomes infinite. When x is positive, the Graphs. 23 curve lies wholly in the first quadrant, approaching positive infinity on the ;ir-axes when y is very small, and positive infinity on the ^-axes when x is very small (Fig. 16). When x is negative, the curve is of the same form and in the third quadrant. Plot the curve y=^—^' Note that if a plane x^ mirror were passed through the ^r-axis perpendicu- lar to the j-axis, the reflection of the curve y ^—^ I ^ in the mirror would be the graph oi y= -• Fig. 16. It has been seen in paragraph 10 that the curves yz=ix^^ etc., touched the ;r-axis, but did not cross it. These curves had equal roots (when jv = o, (jt — i)2 = o), which give .r = i or i. It is in gen- eral true that a curve with equal roots must be tangent to the axis, i,e, must cut the axis in at 24 Supplementary Algebra. least two ^consecutive points. If the equation has in its numerator a binomial factor (x — a) repeated an even number of times, the curve will touch the axis at a, but will not cross at that point. But if the factor be repeated an odd number of times, the curve will touch the axis and cross it also at a [try J = (-^ — I /] . If the binomial is not repeated, the curve crosses the axis at a non-vanishing angle. All curves where the factor repeats are parallel to the axis at the point of intersection. This fact is at once apparent when we remember that the curve is a tangent to the axis at such a point. 12. Maxima and Minima. It may be noticed that many of these curves make a complete turn, as in Fig. 8 or Fig. 1 2 ; that is, there is some point which is higher or lower than any other point of the curve in the immediate vicinity of the point in question. Such turning points are called maximum and minimum points of the function. These points are of considerable geometric value. Problem. Suppose it be required to find the rectangle of greatest area which can be inscribed in an isosceles triangle with an altitude equal to 2 and a base equal to 2. Let ABE (Fig. 17) be the given triangle, DCy one half the required rectangle. Let J be the area of the rectangle and 2 x the base. * Mathematically the word coincident should replace consecutive. The beginner, however, seems more clearly to realize that the intersections are approaching each other indefinitely near if consec- utive is used. Graphs. 25 ~"~~ X 0] Y \ 1 Fig. 18. From the figure, CO : AD-OD : : 2 : i, C0^2{AD-0D\ = 2(i-;r), therefore y^^x — A^x^ (Fig. 18). But this ex- pression is a curve where y represents the area we are trying to find, and may be plotted as any other curve. Plotting the curve, we see that it turns at ;r = |-, j^= I (Fig. 18), and that this point is higher than any other point of the curve. Therefore, the greatest rectangle which can be inscribed in the given triangle is one in which the base and altitude are each i. The rectangle is minimum when x=o and when x= i. Why ? Again : Given a square piece of sheet metal 30 inches on a side, find the side of a square to be 26 Supplementary Algebra. cut out of each corner of the sheet so that the remainder will fold up into a box of maximum volume. Let ABCD (Fig. 19) be the sheet of metal and y the volume of the box. Let x be the side of the square to be cut from each corner. s 7 H^ Fig. 19. Fig. 20. Then from the figure, y = (30 — 2 xfx. Plotting this curve (Fig. 20) we see that it turns twice, once at ;r = 5 and once at x= 1$; also that ;r= 15 is at the lowest part of a bend, while ;r= 5 is at the highest part of the other bend. This shows, then, that the box is greatest when ;r= 5 and least when ;r= 15, for when 15 is taken from each corner no material remains from which to make the box. The graph shows also that X I 2 1352 5 2000 6 1944 8 1568 10 1000 13 208 14 56 15 16 64 Graphs. 27 the greatest box that can be so formed has a volume of 2000, the value of y of the curve, when The turning points of a curve are often difficult to plot because the values of x and y must be taken so closely together. Much labor may often be saved by the following process, called differentia- tion. Only one rule is given here. Should the reader wish rules governing all cases, he will find them in Wells's ** College Algebra," p. 472, and Wells's "Advanced Course in Algebra," p. 527. The following rule holds where j/ equals a rational polynomial in Xy containing no fractions with vari- ables in the denominator : multiply each coefficient by the exponent of x in that term and depress the exponent by one. If the resulting expression is equated to zero and solved for x, the roots thus obtained will be the abscissas of the points where the curve turns. A proof of this will have to be postponed until higher mathematics is reached. Illustration. Use the curve in Fig. 8. j^ = ;r3-;tr2-4;r + 4. (l) Differentiating and equating the result to zero, we have ^:^-2x~4 = 0, (2) whence x=— — ^= 1.5+ or — ,^6+y 3 the turning points of the curve. This result may be verified by examining Fig. 18. These two 28 Supplementary Algebra. Fig. 21. Graphs. 29 curves are plotted on the same axes in Fig. 21, equation (2) being dotted. Problem. An open vessel is to be constructed in the form of a rectangular parallelopiped with a square base, capable of containing 4 cubic inches. What must be the dimensions to require the least amount of material ? Solve by plotting. Ans. Dimensions must be 2 x 2 x i inches. It is often necessary to plot a curve in one vari- able only; for example, ;ir2—;r— 12 = o. In such a case we are really seeking the intersection of two curves, j/ = o and x^—x— 12 = 0, which are plotted as above, y = o being the Hne OX. In such a case the curve is readily plotted if the two equations are equated ; e,g, y^x'^—x— 12. Flot x^ — X — 6 = o; also regard the equation as of the form ax^ -\- bx -{- c = o, and solve by the formula, x = . Note the nature of 2a the radical part of the formula after the substitution is made. Treat x^— x-{-^==o and x^—x-{-6 = o in the same manner. Remember that the places where the curve crosses the ;ir-axis correspond exactly with the algebraic root obtained. This affords an ex- cellent method for illustrating to the student the meaning of imaginary roots. 30 Supplementary Algebra. Short Methods. (To shorten the work in certain classes of fractions involving addition. For review work only.) Rule I. If two fractional numbers having a common numerator and denominators prime to each other are to be added, multiply the sum of the denominators by the common numerator for the numerator of the result; the product of the denominators will be the denominator of the result. Rule II. If two fractional numbers having a common numerator and the denominators prime to each other are to be subtracted, subtract the first denominator from the second, multiply this difference of the denominators by the common numerator for the numerator of the result; the product of the denominators will be the denomi- nator of the result. Illustration : i + * = T^; f + l = 2G + i) = if- 4-i=A; (f-l) = 2(J-i) = ^5- This scheme makes possible a great saving of products in algebraic summations. + ■ (use Rule II), Short Methods. 31 _ 4 , -4 ^J I i_\ (use Rule II), 12 -"(^4-4X^4-1) 2. — "^ ^=-J! —' (Use Rule II.) X—l X—2 ^—3 X—4 — I I . (x-i){x-2)'~'^{x-3)(x-4y whence (^-3)(^-4) = (-^-"i)(^-2); :ir2— 7;ir+ 12 = ;ir2 — 3;ir4- 2. .-. ;r=2l 3. 7= H 7== = 12 : solve for x. X + V ;r2 — I X — -Vx^ — I ( , /% + 7t=V ^2 (use Rule I); \x + V;!:^ — I X — -yx"^ — 1/ 24r 12. ;r2 — (;r2— i) Improper Fractions. The larger the factor, the more the multiplication involved, the greater the liability of error, and the slower the speed. The times demand, in mathe- matical computations, accuracy combined with a reasonable degree of rapidity. 32 Supplementary Algebra. In general, improper fractions have no place in problems involving addition and subtraction. In arithmetic we do not add |, f , f , etc., because the work is cumbersome and the factors large. We reduce these fractions to whole or mixed num- bers, 2f, 2 J, if, add the integers, and then the fractions. We should not think of adding the fractions in their original form. The same prac- tice holds good for algebra. An improper fraction may be defined as a fraction in which the numer- ator is of the same or higher degree than the denominator, both being rational. Examples. X—l x— 2 _ x — 3 X — ^ X—2 X — 2i ^—4 ^-"5 Divide each numerator by its denominator — parenthesis division. I I.I I H I -.= iH ^- i_. X--2 x—i ^—4 ^—5 Collecting, \_X-2 X-l\ \_X-\ X-<>\ (Use Rule II.) —I _ — I (Ar-2)(;r-3)~(;tr-4)(^-5)' {x-^)(x-l) = (x-2){x-z). Short Methods. ^3 Such reduction of improper fractions will in most cases shorten a solution and will reduce the size of the factors ; a binomial numerator generally reduces to a monomial, and a trinomial numerator to a binomial. 2. ^ ^ ^ ^ + 7^+ lO X — 2 x^-{- yx-\- lo Reducing, i + o^,^^,^ - i Collecting, factoring first denominator, 7 r-r^^ r bCCOmOS ; ]• {x -{- i){x -\- 2) X—2 \x-\-2 X—2j' (Use Rule II.) X^—X-^- I . X^-\-X'\- I 3. 1 ; ^2X. X— I X-\' I Reducing, x-^- — 7 + -^ + rT:7 = 2;r, X — 1 X "i" 1 whence, 1 ; — = o. (Apply Rule I.) X—l X-\-l \ irtr J / r.x — o. Problems of the above type are common to all texts on algebra. After the student has solved them by the usual methods, so as to become fa- miliar with the principles involved, he should be given a review where these briefer solutions might be introduced. 34 Supplementary Algebra. Note. A short method should never be introduced early in a subject. Such procedure clouds the pupil's ideas of mathe- matical principles. Simultaneous Quadratics of Homogeneous Form may often be solved as follows : Illustrative Problems. Eliminate the abso- lute terms by multi- plying equation ( i ) by (c) 9f equation (2) by 14, and subtracting. This resulting func- tion of (;r,^)is always capable of being fac- tored, either by in- spection or by quad- ratic factoring and gives two values of x in terms of j. These, when substituted in ( I ) or (2) will produce the required values for J/. X is then easily obtained. This method avoids the substitution of y'=^vx. 2x^ — xy^2S ;i:2 + 2/= 18 (I) (2) i%x^ — gxy = 2S2 l4;r2 + 2872=252 (3) (4) 4;r2 — ()xy — 2872 = (5) (4^+7j/)(^~4J^) = 0. ,\x=-lyor 4y. Substituting in (2), 11/ + 2/ =18. ^^^^-18. 16 y'=¥- .•.J/=±|V2. 16/ +2/= 18. y=i. ± I. Short Methods. 35 Algebraic Problems. The following problems in reduction are of com- mon occurrence in the higher mathematics, and especially in the calculus, where long lists of such forms are found in the work almost daily. They appear most frequently in differentiation and prep- aration for integration ; lack of facility in handling such forms handicaps a pupil and makes him think the calculus a difficult subject. Type Forms. (a + df -{-(a-df=2 (a^ + 6^) ; {a + df-4ad = (a'-df; {a-df-h4ab = {a + df. It is desirable that a student be so familiar with these type forms that he may, by inspection, write down the results of similar operations. a -]- d a — b (7 + V25 -/)2 - (7 - V25 -/)2 = .? Simplify the following : {e^'he-^f-^4, ^6 Supplementary Algebra. ■\/(mn — i)2 + 4 mfif Negative Exponents. Notice all quantities which have negative expo- nents. Multiply both numerator and denominator of the given expression by the product of all such quantities with the sign of the exponents changed. If the expression be integral, regard its denomina- tor as unity. Example (i): Multiply both numerator and denominator by 2 Result = ^ ^ Example (2) : x(i+x^y^-x{i-x^y^. •y/l — X^\I -\- X^ Multiplying by , we obtain Vl —x'^^Y ■\-X^ x^i —x'^ — x^i '\-X^ Short Methods. 37 In like manner simplify : ,-.+^^ fif+f-^-* jrJj/* In all reductions watch for opportunities to factor, especially to remove factors common to several terms. («) Z{x - 2f{2 X -^ if -^ t,{2 X ^ \)\x - 2f = {2X+ lf(x-2f(l8x- II). (b) 3{x-i)\x + 2f + 4(x-if{x+2f = {x-if{x + 2f(7x+s). (c) (a^+x^f- 3 x^(a^+ x^f = y/'^V^{a^- 2 :fi). (^) 8;r(4;r2 - 3)(;r2 + l)*+ 24;r(;^ + I)* = 56;t8(;tr2+i)*. (^) |(;r+l)-^(;p-5)2 + 2(x-5)(;ir+l)* = |(;r+l)-i(^-5)(2;r-l). Problems. 1. Reduce == to the form Vi-;.^ i-^ 38 Supplementary Algebra. a 2. Reduce s-f^^ -- a^ x-\-a x^ x — a 3. Reduce 4. Reduce I V^ to the form ax — x'' ■H^-i^' 5. Reduce m _m m jm to the form X'^—X "" x^-\-x~^ 6. Reduce ' + ■\/x^ — cfi x^ jx^ — a a ^ d^ to the form -\ X ^ x — a 7. Reduce to unity. Short Methods. 39 8. Reduce ax {a^ - x^)i to the form V^2z:^ 9. S^VWK\ If K^ ^^''y-y\ find / — -^ 10. Reduce 2 I Vl — 3;ir — ;tr2 11. Reduce to the form V13 :x^-x^-6 to the form — — 4;^^ 25 I A-2^ y 12. Reduce V2 ax — ;r2 to the form a {x — df 13. 2 tan ;ir 4-(tan ;i:)2 — 3 = o ; find tan ;ir. 14. 2 cos ;r H = 3 ; find cos x. cos;r 40 Supplementary Algebra. 15. h2 cot;r=fVi -f-cot2;r; find cot ;r. cot;r 16. Reduce ; r-^ to the form L_ + I 4 2(l+-sr) I0(l--8r) 5(3 4- 2-8') 17. vS = — ^(i2 + xy - 8. Evaluate 5 when 18. Reduce M 4^ V V2V1-W 2x— 2;rVi —x^ to the form -li-\ — V 19. Reduce _ . 2;r— I ■y/x^ —x— I to the form 2;r— I + 2 ^x^—x— I I V^^^-jF— I 20. Reduce /£~2\i _^ 3 fx^2\'\f 2 Y V;y+2/ 4V;tr+2/ \:rH-2/ W+2/ + 2/ ;i^ — I to the form —^ Short Methods. 21. Reduce [-(^)i^^ X _x ea + e a 2a to the form «(^^-^"^X 4 22. Reduce — r f to the form —y. 23. Reduce X :i— -4/ ,2 41 y to 3-^ + 2/ when y = 4/;r. 24. Reduce to -<=m a^L jr2 -1/2 when ±~-i^2L=i, 42 Supplementary Algebra. 25. Reduce V «VA ah I ^ --^ 26. Reduce ^-T/i/ 3 / i/*\ 'i/^;tr'~^ to ^s 3 ;r^^ when x^ -hj/^ = ^^. 27. Reduce „i\2ll 1 to — ^^ — Y^ when x^ -{-y^ = a^. £^ 2X 28. Reduce 2a 1/2 ^ ? _? to — when ^=:-(^ + ^ «). Short Methods. 43 29. Reduce to 2V;r {2 a — xf 'y/x{2a — xf 30. Reduce 31. ^-^v^:=^= ^^l-"^V -^i), (I) ^ = r— ^-^^ (2) ^1 ^1 Eliminate ;irj between (i) and (2) and get x'^^-y^^ia-Vbf. 32. Reduce \^x-a^x-{-a — \{x'\-df{x — aJ^ X — a (x— 2 (i\\/x 4- d to ^^ — 3 {x-^df 44 Supplementary Algebra. 33. Reduce n(i'\'xY * x""-^ -nii-V xY-^ • x'^ nx"^"^ ^ to 34. Reduce 2(i-a^)-{-4x^ 35. Reduce ^ ^-r^r- to -(tST ■""■ 36. Reduce 37. Reduce ;.(l+^)-^-;t-(l-^)-^ to ifi - ^ 1 . 38. Reduce (l-2;ir'^)^ ^ If 4;tr-2 j^, -" ,^ 2L2^-2;tr+I Vl - 2 W — 4^±?— Ito-^. 2;r2-h2;r+iJ 1+4^ Short Methods. 45 39. Given show that (a+/9)*+(a~/3)*=2>^*. 40. Given a^ (k-af a^ {k-af ; show that x^ +7' = /^^ Wells's Mathematical Series* ALGEBRA. Wells's Essentials of Algebra . • • . , $z.zo A new Algebra for secondary schools. The method of presenting the fundamen- tal topics is more logical than that usually followed. The superiority of the book also appears in its definitions, in the demonstrations and proofs of gen- eral laws, m the arrangement of topics, and in its abundance of examples. Wells's New Higher Algebra . . . . .1.3a The first part of this book is identical with the author's Essentials of Algebra. To this there are added chapters upon advanced topics adequate in scope and difficulty to meet the maximum requirement in elementary algebra. Wells's Academic Algebra . . . • , x.o8 This popular Algebra contains an abundance of carefully selected problems. W^ells's Higher Algebra ...... 1.32 The first half of this book is identical with the corresponding pages of the Aca- demic Algebra. The latter half treats more advanced topics. Wells's College Algebra ...... 1.50 Part II, beginning with Quadratic Equations, bound separately. $1.32. Wells's Advanced Course in Algebra .... $1.5^ A modern and rigorous text-book for colleges and scientific schools. This is the latest and most advanced book in the Wells's series of Algebra. Wells's University Algebra . . • . .1.3a GEOMETRY. Wells's Essentials of Geometry — Plane, 75 cts.; Solid, 75 cts.; Plane and Solid . . . . . . . z.25 This new text offers a practical combination of more desirable qualities than any other Geometry ever published. Wells's Stereoscopic Views of Solid Geometry Figures • .60 Ninety-six cards in manila case. Wells's Elements of Geometry — Revised 1894. — Plane, 75 cts.; Solid, 75 cts.; Plane and Solid . . • • • z.as TRIGONOMETRY. W^ells's New Plane and Spherical Trigonometry • . $1.00 For colleges and technical schools. With Wells's New Six-Place Tables, $1.25. Wells's Complete Trigonometry . . . , . .90 Plane and Spherical. The chapters on plane Trigonometry are identical with those of the book described above. With Tables, $1.08. Wells's New Plane Trigonometry • • • • .60 Being Chapters I-VIII of Wells's Complete Trigonometry. With Tables, 75 cts. Wells's New Six-Place Logarithmic Tables • • • .60 The handsomest tables in print. Large Page. Wells's Four-Place Tables . • • • • .35 ARITHMETIC. Wells's Academic Arithmetic • • • • • $1.00 Correspondence regarding terms for introduction and exchange is cordially invited. D. C. Heath & Co., Publishers, Boston, New York, Chicago Wells's Essentials of Geometry DEVELOPS SKILL IN THE BEST METHOD OF ATTACK IN THE FOLLOWING WAYS It shows the piipii at once the parts of a proposition and the nature of a proof. Sections 36, 39, 40. 11. It iat-'-n 'yrv- ■■■n ;■;;.;.•::! ::,■.;<;;-,;;,-, " •^,.:■H', and gives such aid as will lead the student to see how each proof depends upoa some- thing already given. See pages 17-18. HI. It begins very early to leave something in the proof for the student to do. See section 51 and throughout the book. The converse propositions are generally left to the student. The '-' indirect method '' is used in those first proved and thus gives a hint as to 1k)\v such propositions are disposed of. See page 21. V. .^ Uj:^o :i;:iiu.cr of the simpler propositions, even iu the first book, are left to the student with but a hint. See pages 50, 51, 52, etc. VL iiguics with all auxiliary lines are made ilu e • i^ the student learns why and how they are drawn. VIL Atter Book I the authority for a statement m the proot is r<ot stated, but the quesdon mark used or the section given. D. C. HEATH & CO., Publishers BOSTON NEW YORK CHICAGO MATHEMATICAL MONOGRAPHS ir:SU}-I> UNDER THE GENERAL EDITORSHIP OF WEBSTER WELLS, S.B. Profiwmr of Matluyncdl a in the ^fai::■'■■■ / ;/.: :'■.; Insiiiide of Technology. It is the purpose of this series to make direct contribution to the resources of teachers of mathematics, by presenting freshly written and interesting monographs upon the history, theory, subject-matter, and methods of teaching both elemen- tary and advanced topics. The first numbers are as follows : — 1. FAMOUS GEOMETRICAL THEOREMS AND PROBLEMS AND THEIR HISTORY. By William W. Rupert, C.E. u TJie Greek Geometers, iu The Pythagorean Proposition. 2. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert. ii. The Pythagorean Proposition (concluded), iii. Squaring the Circle.' 3. FAMOUS GEOMETRICAL THEOREMS. By W^illlam W. RLtptRT. jv. TnsrctK vn vA au Angle. Y. The Area of a Triangle in Ternis of its Sides. 4. FAMOUS GEOMETRICAL THEOREMS. By William W, Rupert. vj. The Duplication of the Cube. vii. Mathematical. Inscription upon the 'i"i>i.i. u.,;r: of LiKlolph Van Ccnlcn. 5. OJ^ TEACHING GEOMETRY. By Florence Milner. 6. GRAPHS. By rrofr-ssor R. J. Ally, Indiana University. 7. FACXORIK^G. By l^ofessor W,i^:b.ster Wells. PRICE, lo CENTS EACH D. C HEATIi & CO., Publishers, Boston. New York. Chicago (GAyLAMOUNrl PAMPHUT BINOEk Manufaclutid by ;gaylordbros.Uc.3 Syracus*, N. Y. Stockton, Calif. __~^^j^ojectto immediate recall. "PR 5 msm 27Nov'57GC lOV IS flE7 22fab'60J(J REC'D CD 2l-100m-l,'54(1887sl6) 476